CAME 02 Tournaments

Learning Objective:

• we are learning to solve problems by being systematic
• we are learning to use algebra to generalise

Investing:

• The whole point this lesson is to help you learn why thinking logically can help you arrive at a correct solution more quickly
• A useful advantage of using algebra is that when you solve one problem you solve many

Preparing:

• What do you know about the Champion's League Group Stages?
  
  
• Here is an excerpt from the Champion's League Rule Book:

• If you're interested, you can download the Champion's League Rule Book here.

• Notice that there are eight groups of four teams.
• Within each group the four teams all play each other twice: once at home and once away.
• Here's a group from the 2013-14 Champion's League:

• Now I'm an Arsenal fan, so this table looks pretty good to me so far,

• but what I'm wondering is...
  
  
• ... how many games will be played in total within this group?
  
  
• How can you b e sure you've got them all?
• What method did you use to find them?

Discovering:

• Maybe you need some help to get started?
• Here is a bit more from the rule book. It's not complete.

• Unfortunately the rest of the fixtures were on the next page, so some games are missing.
• Finish off the fixture list .

• Investigate...
• what if UEFA decided to change the rules next year and put 5 teams into each group?
• How many games in each group now?
  
  
• what if they put 6 teams into each group instead?

• What is your way of finding how many games are played by any number of teams?
• Explain to your learning partner how you work it out.
• Compare your strategies.

• Whose method is better? Why?

Modeling:

• Here are some examples of people getting it right:
• Here are some examples of people getting it wrong in typical ways:

Extending:

• The English Premiership has 20 teams:

• You know why I had to show you that don't you?
• So how many games this season in the whole Premier League competition?

Explaining:

• How can you prove you've got the right number of matches?

Generalising:

• Okay. Now for the cool bit...
• Suppose there are some number, n, teams in a league.
• How would you work out how many matches they play (assuming they all play each other once at home and once away)?

Developing:

• In the Champion's League, once the group stages end, some of the best teams then play in a knock-out tournament to get a winner.
• If there were only four teams in the knock-out, how many games would be needed to get a knock-out winner?
• What if there are eight teams?
• What if there are sixteen?
• What about other numbers of teams?
  
  
• If some number, n, teams are in the knock-out stage, how many games are needed to get to the winner?
  
  
• Why?
  
  
• What if they play two legs to each knock-out match, one at home and one away?
  
  
• What if there are two legs to all the knock-out matches except the the final which, instead of being played over two legs home and away, is played in a single game at a neutral ground?
• (This is the format of the Champion's league)

Assessing:

• Explore the number of games in other leagues and competitions.
  • Write a report about the arrangements in a sport of your choice.

• Suggested titles might include "the road to the Superbowl" in American Football and "chasing down the Heineken Trophy" in Rugby Union.
  
  
• Can you find a competition with an interesting or unusual group and/or knock-out structure
  • How does it work?

• Download the rules for the Heineken Cup
  • Read how the cup group and knock-out stage works.
  • Explain, using your mathematical understanding, some plausible reason(s) why the Cup organisers decided to have six leagues of four teams and a set of quarter-finals with the six group winners alongside the highest scoring two second-placed teams rather than four groups of six teams with winners and runners-up going through from each group.
  • How else might they have found a winner from the 24 teams?

Developing:

• What this might have to do with:
  • handshakes
  • computer networks
  • team dynamics
  • permutations, combinations and probability
  • menus